Nsryjdtyk

up vote 6 down vote favorite 1 Can homological algebra be done with nonabelian groups? In particular, can homology or cohomology be defined on chain complexes of nonabelian groups? I know that Abelian categories are the choice settings for homological algebra, but the notions of kernel and cokernel (which seem to be all that is necessary to define homology) seem to make sense for nonabelian groups as well, if we define $operatorname{coker}(f : G to H)$ to be the quotient of $H$ by the normal subgroup generated by $operatorname{im} f$ . For example, given a sequence of nonabelian groups $$ dotsb to C_3 xrightarrow{partial_3} C_2 xrightarrow{partial_2} C_1 xrightarrow{partial_1} C_0 to 0 $$ with $partial_{n} circ partial_{n+1} = 0$ , is it useful to define the homology groups $H_n(C_bullet) = ker partial_n/N$ , where $N$ is t...